Left Termination of the query pattern rev_in_2(a, g) w.r.t. the given Prolog program could not be shown:



Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

Clauses:

rev([], []).
rev(.(X, Xs), Ys) :- ','(rev(Xs, Zs), app(Zs, .(X, []), Ys)).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Queries:

rev(a,g).

We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2)  =  rev_in
.(x1, x2)  =  .(x2)
U1(x1, x2, x3, x4)  =  U1(x4)
[]  =  []
rev_out(x1, x2)  =  rev_out(x1, x2)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
app_in(x1, x2, x3)  =  app_in(x1, x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
app_out(x1, x2, x3)  =  app_out(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
PiTRS
      ↳ DependencyPairsProof
  ↳ PrologToPiTRSProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2)  =  rev_in
.(x1, x2)  =  .(x2)
U1(x1, x2, x3, x4)  =  U1(x4)
[]  =  []
rev_out(x1, x2)  =  rev_out(x1, x2)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
app_in(x1, x2, x3)  =  app_in(x1, x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
app_out(x1, x2, x3)  =  app_out(x3)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

REV_IN(.(X, Xs), Ys) → U11(X, Xs, Ys, rev_in(Xs, Zs))
REV_IN(.(X, Xs), Ys) → REV_IN(Xs, Zs)
U11(X, Xs, Ys, rev_out(Xs, Zs)) → U21(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
U11(X, Xs, Ys, rev_out(Xs, Zs)) → APP_IN(Zs, .(X, []), Ys)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U31(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2)  =  rev_in
.(x1, x2)  =  .(x2)
U1(x1, x2, x3, x4)  =  U1(x4)
[]  =  []
rev_out(x1, x2)  =  rev_out(x1, x2)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
app_in(x1, x2, x3)  =  app_in(x1, x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
app_out(x1, x2, x3)  =  app_out(x3)
REV_IN(x1, x2)  =  REV_IN
U31(x1, x2, x3, x4, x5)  =  U31(x5)
U21(x1, x2, x3, x4)  =  U21(x2, x4)
APP_IN(x1, x2, x3)  =  APP_IN(x1, x2)
U11(x1, x2, x3, x4)  =  U11(x4)

We have to consider all (P,R,Pi)-chains

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
PiDP
          ↳ DependencyGraphProof
  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

REV_IN(.(X, Xs), Ys) → U11(X, Xs, Ys, rev_in(Xs, Zs))
REV_IN(.(X, Xs), Ys) → REV_IN(Xs, Zs)
U11(X, Xs, Ys, rev_out(Xs, Zs)) → U21(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
U11(X, Xs, Ys, rev_out(Xs, Zs)) → APP_IN(Zs, .(X, []), Ys)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U31(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2)  =  rev_in
.(x1, x2)  =  .(x2)
U1(x1, x2, x3, x4)  =  U1(x4)
[]  =  []
rev_out(x1, x2)  =  rev_out(x1, x2)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
app_in(x1, x2, x3)  =  app_in(x1, x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
app_out(x1, x2, x3)  =  app_out(x3)
REV_IN(x1, x2)  =  REV_IN
U31(x1, x2, x3, x4, x5)  =  U31(x5)
U21(x1, x2, x3, x4)  =  U21(x2, x4)
APP_IN(x1, x2, x3)  =  APP_IN(x1, x2)
U11(x1, x2, x3, x4)  =  U11(x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 4 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
PiDP
                ↳ UsableRulesProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2)  =  rev_in
.(x1, x2)  =  .(x2)
U1(x1, x2, x3, x4)  =  U1(x4)
[]  =  []
rev_out(x1, x2)  =  rev_out(x1, x2)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
app_in(x1, x2, x3)  =  app_in(x1, x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
app_out(x1, x2, x3)  =  app_out(x3)
APP_IN(x1, x2, x3)  =  APP_IN(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP_IN(x1, x2, x3)  =  APP_IN(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

APP_IN(.(Xs), Ys) → APP_IN(Xs, Ys)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
PiDP
                ↳ UsableRulesProof
  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

REV_IN(.(X, Xs), Ys) → REV_IN(Xs, Zs)

The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2)  =  rev_in
.(x1, x2)  =  .(x2)
U1(x1, x2, x3, x4)  =  U1(x4)
[]  =  []
rev_out(x1, x2)  =  rev_out(x1, x2)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
app_in(x1, x2, x3)  =  app_in(x1, x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
app_out(x1, x2, x3)  =  app_out(x3)
REV_IN(x1, x2)  =  REV_IN

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

REV_IN(.(X, Xs), Ys) → REV_IN(Xs, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
REV_IN(x1, x2)  =  REV_IN

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ NonTerminationProof
  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

REV_INREV_IN

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

REV_INREV_IN

The TRS R consists of the following rules:none


s = REV_IN evaluates to t =REV_IN

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from REV_IN to REV_IN.




We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2)  =  rev_in
.(x1, x2)  =  .(x2)
U1(x1, x2, x3, x4)  =  U1(x4)
[]  =  []
rev_out(x1, x2)  =  rev_out(x1, x2)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
app_in(x1, x2, x3)  =  app_in(x1, x2)
U3(x1, x2, x3, x4, x5)  =  U3(x2, x3, x5)
app_out(x1, x2, x3)  =  app_out(x1, x2, x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
PiTRS
      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2)  =  rev_in
.(x1, x2)  =  .(x2)
U1(x1, x2, x3, x4)  =  U1(x4)
[]  =  []
rev_out(x1, x2)  =  rev_out(x1, x2)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
app_in(x1, x2, x3)  =  app_in(x1, x2)
U3(x1, x2, x3, x4, x5)  =  U3(x2, x3, x5)
app_out(x1, x2, x3)  =  app_out(x1, x2, x3)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

REV_IN(.(X, Xs), Ys) → U11(X, Xs, Ys, rev_in(Xs, Zs))
REV_IN(.(X, Xs), Ys) → REV_IN(Xs, Zs)
U11(X, Xs, Ys, rev_out(Xs, Zs)) → U21(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
U11(X, Xs, Ys, rev_out(Xs, Zs)) → APP_IN(Zs, .(X, []), Ys)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U31(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2)  =  rev_in
.(x1, x2)  =  .(x2)
U1(x1, x2, x3, x4)  =  U1(x4)
[]  =  []
rev_out(x1, x2)  =  rev_out(x1, x2)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
app_in(x1, x2, x3)  =  app_in(x1, x2)
U3(x1, x2, x3, x4, x5)  =  U3(x2, x3, x5)
app_out(x1, x2, x3)  =  app_out(x1, x2, x3)
REV_IN(x1, x2)  =  REV_IN
U31(x1, x2, x3, x4, x5)  =  U31(x2, x3, x5)
U21(x1, x2, x3, x4)  =  U21(x2, x4)
APP_IN(x1, x2, x3)  =  APP_IN(x1, x2)
U11(x1, x2, x3, x4)  =  U11(x4)

We have to consider all (P,R,Pi)-chains

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
PiDP
          ↳ DependencyGraphProof

Pi DP problem:
The TRS P consists of the following rules:

REV_IN(.(X, Xs), Ys) → U11(X, Xs, Ys, rev_in(Xs, Zs))
REV_IN(.(X, Xs), Ys) → REV_IN(Xs, Zs)
U11(X, Xs, Ys, rev_out(Xs, Zs)) → U21(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
U11(X, Xs, Ys, rev_out(Xs, Zs)) → APP_IN(Zs, .(X, []), Ys)
APP_IN(.(X, Xs), Ys, .(X, Zs)) → U31(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2)  =  rev_in
.(x1, x2)  =  .(x2)
U1(x1, x2, x3, x4)  =  U1(x4)
[]  =  []
rev_out(x1, x2)  =  rev_out(x1, x2)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
app_in(x1, x2, x3)  =  app_in(x1, x2)
U3(x1, x2, x3, x4, x5)  =  U3(x2, x3, x5)
app_out(x1, x2, x3)  =  app_out(x1, x2, x3)
REV_IN(x1, x2)  =  REV_IN
U31(x1, x2, x3, x4, x5)  =  U31(x2, x3, x5)
U21(x1, x2, x3, x4)  =  U21(x2, x4)
APP_IN(x1, x2, x3)  =  APP_IN(x1, x2)
U11(x1, x2, x3, x4)  =  U11(x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 4 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
PiDP
                ↳ UsableRulesProof
              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2)  =  rev_in
.(x1, x2)  =  .(x2)
U1(x1, x2, x3, x4)  =  U1(x4)
[]  =  []
rev_out(x1, x2)  =  rev_out(x1, x2)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
app_in(x1, x2, x3)  =  app_in(x1, x2)
U3(x1, x2, x3, x4, x5)  =  U3(x2, x3, x5)
app_out(x1, x2, x3)  =  app_out(x1, x2, x3)
APP_IN(x1, x2, x3)  =  APP_IN(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

APP_IN(.(X, Xs), Ys, .(X, Zs)) → APP_IN(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APP_IN(x1, x2, x3)  =  APP_IN(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP

Q DP problem:
The TRS P consists of the following rules:

APP_IN(.(Xs), Ys) → APP_IN(Xs, Ys)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
PiDP
                ↳ UsableRulesProof

Pi DP problem:
The TRS P consists of the following rules:

REV_IN(.(X, Xs), Ys) → REV_IN(Xs, Zs)

The TRS R consists of the following rules:

rev_in(.(X, Xs), Ys) → U1(X, Xs, Ys, rev_in(Xs, Zs))
rev_in([], []) → rev_out([], [])
U1(X, Xs, Ys, rev_out(Xs, Zs)) → U2(X, Xs, Ys, app_in(Zs, .(X, []), Ys))
app_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, app_in(Xs, Ys, Zs))
app_in([], X, X) → app_out([], X, X)
U3(X, Xs, Ys, Zs, app_out(Xs, Ys, Zs)) → app_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Xs, Ys, app_out(Zs, .(X, []), Ys)) → rev_out(.(X, Xs), Ys)

The argument filtering Pi contains the following mapping:
rev_in(x1, x2)  =  rev_in
.(x1, x2)  =  .(x2)
U1(x1, x2, x3, x4)  =  U1(x4)
[]  =  []
rev_out(x1, x2)  =  rev_out(x1, x2)
U2(x1, x2, x3, x4)  =  U2(x2, x4)
app_in(x1, x2, x3)  =  app_in(x1, x2)
U3(x1, x2, x3, x4, x5)  =  U3(x2, x3, x5)
app_out(x1, x2, x3)  =  app_out(x1, x2, x3)
REV_IN(x1, x2)  =  REV_IN

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof

Pi DP problem:
The TRS P consists of the following rules:

REV_IN(.(X, Xs), Ys) → REV_IN(Xs, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
REV_IN(x1, x2)  =  REV_IN

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ NonTerminationProof

Q DP problem:
The TRS P consists of the following rules:

REV_INREV_IN

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

REV_INREV_IN

The TRS R consists of the following rules:none


s = REV_IN evaluates to t =REV_IN

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from REV_IN to REV_IN.